I DDWFTTW: Looking for the least confusing explanation

I have tried to find the least confusing (or rather, the most 'deconfusing') explanation for why it is possible for a cart with a prop to travel directly downwind faster than the wind. So far this is what I've come up with:

Initial discussion

At first glance, a cart travelling down faster than the wind would experience a headwind. The drag of the headwind would slow down the cart. However, if that cart also has spinning blades, the back side of the blades could run into the air, such that the pressure on the blades is from behind, not in front. So this would have the effect of applying a forward pressure on the vehicle which would force the cart, prop, bearings, etc. to move forward.

Because the bearings hold the axle, the increased translational velocity of bearings relative to the ground would cause the wheels and axles to spin. So at the expense of some of the gained translational kinetic energy of the cart and bearings, the wheels and axles will gain rotational kinetic energy in addition to their own translational kinetic energy. Because the prop is connected to the axle via a belt, the prop must then recover rotational kinetic energy at the expense of the rotational and translational kinetic energies of the rest of the cart (as well at the expense of some of the prop's own translational kinetic energy). This gain of rotational kinetic energy by the prop must overcome the loss of rotational kinetic energy due to the drag by the air that it slices through.

Revealing the hidden assumption

When we say the cart is travelling directly downwind faster than the wind, we are saying that both the cart and the wind have a speed, but one is greater - relative to something. Strictly speaking, the norm of the velocity difference between the cart and this something is greater than the norm of the velocity difference between the air and this something.

In the initial discussion, we choose this something to be our ground. Then we say the cart is travelling directly down the wind faster than the wind as a result of work done on it (relative to the ground) and that kinetic energy of the wind (relative to the ground) is extracted upon by the cart to provide this work.

Forces or work, which is more important?

Ignoring relativity, the forces do not change with the frame of reference, but the displacements do. Therefore the work done on or done by the air, ground, cart, the parts, etc. is dependent on the inertial frame of the independent observer. So if we choose an observer independent of the ground, then work may be done by the air on the cart, or vice versa, and work may be done by the ground on the cart, or vice versa.

However, if by displacements we meant displacements relative to the initial rest frame of each material entity and not displacements relative to an arbitrary inertial observer, then given a reasonable assumption that the ground has a very large inertia relative to the cart and the air, the ground may be seen as receiving no work upon it, according to its unchanging rest frame. The cart may be receiving a force according to its frame, but the rate at which work is done on the cart it is ill defined in this frame unless an initial rest frame for the cart can be chosen. If this could be done for both the cart and the air, then one may say the air does work on the cart (or vice versa), depending on the initial "rest" frames for the air and the cart. Lacking specification of these initial rest frames, then the transfer of energy between the air and the cart and between the ground and the cart is frame-dependent.

So it is not so much that the wind does work on the cart, or the ground does work on the cart, or whatever combination one so desires, since that is the arbitrary result of choosing a frame of reference. In the end, we are dealing with differences in velocity and their norms (i.e. speeds relative to ground), and therefore differences in momentum and their derivatives with time (i.e. forces). Therefore, the deciding factor that makes DDWFTTW possible are the forces on the bodies and not work done between them.

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.

The easiest frame of reference is probably the cart's frame of reference. Assume the cart is not accelerating, so that its frame of reference is inertial. When the car is moving fast enough to experience an apparent headwind (or zero wind) from its frame of reference, the ground is moving faster. There are two sets of Newton third law pairs: the wheels exert a forwards force onto the ground coexistent with the ground exerting a backwards force onto the wheels which in turn drive the propeller which exerts a backwards force onto the air coexistent with the air exerting a forwards force onto the propeller. Since the ground is moving faster than the apparent wind, the cart exploits the situation using an effective reduction in gearing from wheels to propeller, which divides the speed and multiplies the force. The propeller generates a greater force but at a lesser speed, exploiting the fact that the propeller interacts with the slower moving apparent headwind. There are losses in the system (aerodynamic drag, rolling resistance, drivetrain losses, ... ) but a DDWFTTW vechicle named "Blackbird" was efficient enough to achieve a downwind speed about 2.8 times the true wind speed (about 28 mph in a 10 mph wind).

From the carts frame of reference, energy is extracted by slowing down the speed of the earth's surface (with respect to the cart). The earth is massive so the change in speed is extremely tiny. The extracted energy, minus losses, is added to the apparent headwind by the propeller.

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From an inertial frame of reference tied to the surface of the earth at a moment in time, the cart's propeller slows down the true wind, extracting energy from the true wind, and from that inertial frame of reference, after losses, adds some energy to the earth by speeding up the movement of the earth's surface by a very tiny amount.

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.

Some powers are frame independent:
- power transmitted in a shaft : angular velocity * torque (across inertial frames)
- power dissipated / generated / consumed at the interface of two moving bodies (based on their relative velocity)

Once you understand how yachts tack downwind with velocity made good greater than windspeed, DDWFTTW is almost trivial ...

I never liked the tacking analogy. A DDWFTTW cart could use any efficient thrust generating mechanism driven by the wheels. Propellers are used because of efficiency, not because of a similarity to tacking.

Has anyone investigated if its necessary to have the wheels or water prop power the rotor?

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it. Ok so there are some engineering challenges but the principle would seem to be ok.

You can argue that not all of the rig is going directly down wind but not all of the cart discussed above does either.

Has anyone investigated if its necessary to have the wheels or water prop power the rotor?

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it. Ok so there are some engineering challenges but the principle would seem to be ok.

You can argue that not all of the rig is going directly down wind but not all of the cart discussed above does either.

This is the reason I don't like the tacking analogy for a DDWFTTW cart. With the pair of land yachts, the ground is opposing the apparent crosswind experienced by each land yacht, which is a function of the true wind times sin(heading with respect to wind), and independent of the land yacht's speed.

In theory, a DDWFTTW cart could be made with no parts that move perpendicular to the true wind. For example, the wheels could drive an enclosed paddle wheel (axis parallel to wheel axis) with vents located front (intake) and back (output) so that the vented paddle wheel generates thrust (similar to a vane pump).

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it.

Yes, and the lateral movement of the yachts would be powered by their keels, just like the rotation of the propeller is powered by the wheels. Note that the aerodynamic force on the yachts opposes their lateral movement, just like the aerodynamic torque at the prop opposes its rotation.

I guess you mean that the surface provides the force that drives the yachts laterally (perpendicular to true wind). Just like the DDWFTTW wheels provide the torque, that drives the propeller blades laterally.

But you can make it even simpler:

In the surface frame the interaction with the surface (via keel / wheels) is just a constraint, which does no work, but merely constrains the airfoils (yacht sails / prop blades ) to a certain path.

I guess you mean that the surface provides the force that drives the yachts laterally (perpendicular to true wind). Just like the DDWFTTW wheels provide the torque, that drives the propeller blades laterally.

Ignoring drag, the force from the surface does no work because the force is perpendicular to the path of the land yacht. As I posted before, it opposes the apparent crosswind which = true wind times sin(heading with respect to true wind), allowing the apparent crosswind to be diverted by the sail to produce thrust to drive the land yacht forward.

In a DDWFTTW cart, the ground force is in the opposite direction of the path of the cart, performing "negative" work on the cart to drive the propeller, which combined with the tailwind, performs "positive" work on the cart.

Ignoring drag, the force from the surface does no work because the force is perpendicular to the path of the land yacht. As I posted before, it opposes the apparent crosswind which = true wind times sin(heading with respect to true wind), allowing the apparent crosswind to be diverted by the sail to produce thrust to drive the land yacht forward.

In a DDWFTTW cart, the ground force is in the opposite direction of the path of the cart, performing "negative" work on the cart to drive the propeller, which combined with the tailwind, performs "positive" work on the cart.

You really should take the advice from the OP:

If one insists to use an work-based or power-based explanation, they should clearly specify the frame of reference in which the kinetic energies are defined, and if they are going to analyze the problem from different frames of reference, they should make it clear that kinetic energies and powers are not the same between different frames of reference. If they jump between different frames of reference when thinking about the DDWFTTW in terms of kinetic energies and powers, they will likely confuse themselves if they are not careful.

The key point I was making was that in the land yacht case, the surface force is perpendicular to the velocity of the land yacht (ignoring drag), while in the DDWFTTW case, the surface force is in the opposite direction of the velocity of a DDWFTTW cart

The tacking analogy doesn't have one tacking yacht, but two connected yachts:

It's known that some racing sailing boats or land yachts can go down wind faster than the wind by tacking. So in theory you could arrange for two of those (on opposite tacks) to tow a light weight barge. The barge could go directly down wind while the two boats tack back and forth in front of it.

The net force from the surface on the whole contraption points backwards, just like for the DDWFTTW cart.

No, the path is frame dependent, and so is the work done by a force acting along that path.

It doesn't matter that the path is frame dependent, only that the force is always perpendicular to the path (perpendicular to the land yacht's velocity), regardless of the frame of reference. It's similar to the situation where a centripetal force never performs any work, regardless of the frame of reference.

From the sail boat or land yacht frame of reference, the lateral force from the keel or wheels is part of a Newton third law pair. The lateral force from the keel or wheel is opposed by the lateral force related to diversion of the apparent wind, and the net lateral force is zero (otherwise the sail boat or land yacht would be accelerating laterally). Since the net force from this Newton third law pair of forces is zero, it doesn't matter which frame of reference is used, the net force remains zero.

From the sail boat or land yacht frame of reference, what drives the sail boat or land yacht forward is the diversion of the apparent crosswind in a direction aft of the sail boat or land yacht. Again a Newton third law pair, the sail exerts a backwards force on the apparent crosswind, and the affected air exerts a forward force on the sail. As stated before, the lateral forces cancel. Diversion of the apparent headwind contributes to part of the lateral force and also somewhat reduces the net forward force from the sail (similar to lift induced drag). From a ground frame of reference, a component of the flow diverted from the sail is upwind, which reduces the true wind speed (extracting energy from the true wind).

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As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart, but if the propeller was replaced by a paddle wheel like thrust generator, there would be no component of movement perpendicular to the wind.

As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart,

Yes, and in the frame of the twin-yacht-vehicle the lateral movement of the sails is powered by the keels, just like the rotation of the propeller blades is powered by the wheels in the DDWFTTW-cart frame.

Yes, and in the frame of the twin-yacht-vehicle the lateral movement of the sails is powered by the keels, just like the rotation of the propeller blades is powered by the wheels in the DDWFTTW-cart frame.

The lateral movement of the sails would be powered by the lateral component of forces from the keels. The true-wind component of forces from the keels would be an upwind force.

As for the two land yacht contraption, an analogy could be made comparing the sails to a rotating propeller on a DDWFTTW cart, but if the propeller was replaced by a paddle wheel like thrust generator, there would be no component of movement perpendicular to the wind.

I have no idea why you even find this relevant, but a rotating paddle wheel does have parts moving perpendicular to the driving direction (up / down).

There would be movement perpendicular to the true wind, but not interaction. The parts motion would include perpendicular motion, but in the idealized vehicle, there would be no net interaction with the two media (air and ground or air and water) during any motion perpendicular to the true wind. For example, the wheels driven by the ground in a conventional DDWFTTW vehicle rotate, but there are no net interactions or forces between wheel and ground that are perpendicular to the true wind. My point was that a idealized thrust generator could be used to interact with the air, so that are no net interactions or forces between thrust generator and air that are perpendicular to the true wind, and that an idealized DDWFTTW vehicle would not require any net interaction or forces between the vehicle and the two media that are perpendicular to the true wind. It's just an issue of efficiency.

The key point of DDWFTTW or DUWFTTW is that tacking is not required.

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Getting back to the least confusing explanation of a a vehicle that interacts with two media (for this case air and ground) with a relative velocity (true wind speed from the ground frame of reference) ... One explanation focuses on the effective gearing used to reduce the speed and increase the force from the faster moving media (ground if moving DDWFTTW, air if moving DUWFTTW) to be applied to the slower moving media, which is used in the articles you linked to that discuss the process from a power perspective, which are similar to Mark Drela's analysis (link below). Another explanation uses tacking as an analogy, which is somewhat different in the case of a single land yacht (which is why I don't like tacking analogy). However, in the case of a two land yacht vehicle, the same effective gearing effect is involved, the net force on the vehicle from the sails that interact with the air is downwind and the net force on the vehicle from the wheels (acting like keels) that interact with the ground is upwind, and from the vehicles frame of reference, the gearing is used to reduce the speed and increase the force from the faster moving media.